relational beta-module in nLab

Relational -modules

Relational β\beta-modules

Idea

One of my early Honours students at Macquarie University baffled his proposed Queensland graduate studies supervisor who asked whether the student knew the definition of a topological space. The aspiring researcher on dynamical systems answered positively: “Yes, it is a relational β\beta-module!” I received quite a bit of flak from colleagues concerning that one; but the student Peter Kloeden went on to become a full professor of mathematics in Australia then Germany. —Ross Street, in An Australian conspectus of higher categories?

In 1970, Michael Barr gave an abstract definition of topological space based on a notion of convergence of ultrafilters (building on work by Ernest Manes on compact Hausdorff spaces). Succinctly, Barr defined topological spaces as ‘relational β\beta-modules’. It was subsequently realized that this was a special case of the notion of generalized multicategory. Here we unpack this definition and examine its properties.

The correctness of this definition (in the sense of matching Bourbaki's definition) is equivalent to the ultrafilter principle (UPUP). However, the definition can be treated on its own, even in a context without UPUP. So we also consider the properties of relational β\beta-modules when these might not match Bourbaki’s open set definition of topological spaces.

Definitions

Abstract description

If SS is a set, let βS\beta{S} be the set of ultrafilters on SS. This set is canonically identified with the set of Boolean algebra homomorphisms

P(S)→2,P(S) \to \mathbf{2},

from the power set of SS to 2\mathbf{2}, the unique Boolean algebra with two elements. The 2-element set carries a dualizing object structure that induces an evident adjoint pair

(Set→PBool op)⊣(Bool op→hom(−,2)Set)(Set \stackrel{P}{\to} Bool^{op}) \; \dashv \; (Bool^{op} \stackrel{\hom(-, \mathbf{2})}{\to} Set)

so that the composite functor β=hom(P−,2):Set→Set\beta = \hom(P-, \mathbf{2}): Set \to Set carries a monad structure.

The functor β:Set→Set\beta : Set \to Set extends to Rel as follows: given a binary relation r:X→Yr\colon X \to Y, written as a subobject in SetSet

R→⟨π 1,π 2⟩X×Y,R \stackrel{\langle \pi_1, \pi_2 \rangle}{\to} X \times Y,

we define β(r):β(X)→β(Y)\beta(r): \beta(X) \to \beta(Y) to be the relation obtained by taking the image of ⟨β(π 1),β(π 2)⟩:β(R)→β(X)×β(Y)\langle \beta(\pi_1), \beta(\pi_2) \rangle: \beta(R) \to \beta(X) \times \beta(Y). It turns out, although it is by no means obvious, that β\beta is according to this definition a strict functor on RelRel.

The monad structure on β:Set→Set\beta: Set \to Set, given by a unit u:1→βu: 1 \to \beta and multiplication m:ββ→βm: \beta \beta \to \beta, extends not to a strict monad on RelRel, but rather one where the transformations u,mu, m are op-lax in the sense of there being inequalities

X →u X βX ββX →m X βX r↓ ≤ ↓β(r) ββ(r)↓ ≤ ↓β(r) Y →u Y βY ββY →m Y βY\array{ X & \stackrel{u_X}{\to} & \beta X & & & & & & \beta \beta X & \stackrel{m_X}{\to} & \beta X \\ \mathllap{r} \downarrow & \leq & \downarrow \mathrlap{\beta(r)} & & & & & & \mathllap{\beta \beta (r)} \downarrow & \leq & \downarrow \mathrlap{\beta(r)} \\ Y & \underset{u_Y}{\to} & \beta Y & & & & & & \beta \beta Y & \underset{m_Y}{\to} & \beta Y }

(while of course the monad associativity and unit conditions remain as equations: hold on the nose).

Then a relational β\beta-module is a lax algebra (module) of β\beta on the 2-poset RelRel. In other words, a set SS equipped with a relation ξ:βS→S\xi: \beta S \to S such that the following inequalities hold:

(1)S →u S βS ββS →m S βS 1 S↘≤ ↓ξ β(ξ)↓ ≤ ↓ξ S βS →ξ S \array{ S & \stackrel{u_S}{\to} & \beta S & & & & & & \beta \beta S & \stackrel{m_S}{\to} & \beta S \\ & \mathllap{1_S} \searrow \; \leq & \downarrow \mathrlap{\xi} & & & & & & \mathllap{\beta(\xi)} \downarrow & \leq & \downarrow \mathrlap{\xi} \\ & & S & & & & & & \beta S & \underset{\xi}{\to} & S }

Arguably, it is better to consider RelRel as a proarrow equipment in this construction, in order to accommodate continuous functions between topological spaces (not continuous relations!) as the appropriate abstract notion of morphism between relational β\beta-modules. We touch on this below, but for a much wider context, see generalized multicategory.

Bridge to a concrete description

A relational β\beta-module is a set SS and a binary relation ξ:βS→S\xi: \beta S \to S between ultrafilters on SS and elements of SS that satisfy the conditions (1) . For F∈βSF \in \beta S and x∈Sx \in S, we write F⇝ ξxF \rightsquigarrow_\xi x if (F,x)(F, x) satisfies the relation ξ\xi, or often just F⇝xF \rightsquigarrow x if the relation is clear. We pronounce this by saying “the ultrafilter FF converges to the point xx”, so that ξ\xi plays the role of “notion of convergence”.

Preliminary to explaining the conditions (1), we first set up a Galois connection between ξ∈Rel(βS,S)\xi \in Rel(\beta S, S) and subsets 𝒞∈PP(S)\mathcal{C} \in P P(S), so that fixed points on the PP(S)P P(S) side are exactly topologies on SS, and fixed points on the other side are (as we show below) lax β\beta-module structures on SS. The Galois connection would then of course restrict to a Galois correspondence between topologies and lax module structures.

Recall that each topology 𝒪⊆P(S)\mathcal{O} \subseteq P(S) induces a notion of convergence where F⇝xF \rightsquigarrow x means N x⊆FN_x \subseteq F (FF contains the filter of neighborhoods of xx). Accordingly, for general 𝒞⊆P(S)\mathcal{C} \subseteq P(S), define the relation conv(𝒞)=ξ:βS→Sconv(\mathcal{C}) = \xi: \beta S \to S by

F⇝ ξx⇔(∀ U:P(S))U∈𝒞∧x∈U⇒U∈F.F \rightsquigarrow_\xi x \;\;\; \Leftrightarrow \;\;\; (\forall_{U: P(S)})\; U \in \mathcal{C} \; \wedge \; x \in U \; \Rightarrow \; U \in F.

Conversely, a topology 𝒪\mathcal{O} can be retrieved from its notion of convergence: under the ultrafilter principle, the neighborhood filter of a point xx is just the intersection of all ultrafilters containing it (hence all FF such that F⇝xF \rightsquigarrow x), and then a set is open if it is a neighborhood of all of its elements. Accordingly, for general “notions of convergence” ξ∈Rel(βS,S)\xi \in Rel(\beta S, S), we define a collection τ(ξ)⊆P(S)\tau(\xi) \subseteq P(S) by

τ(ξ)≔{U⊆S:(∀ F:βS,x:S)(x∈U∧F⇝ ξx)⇒U∈F}.\tau(\xi) \coloneqq \{U \subseteq S: \; (\forall_{F: \beta S, x: S})\; (x \in U \; \wedge\; F \rightsquigarrow_\xi x) \Rightarrow U \in F\}.

Propositions , , and more or less show that a topological space (S,𝒪)(S, \mathcal{O}) is a particular type of pseudotopological space:

All that remains is to check is:

The unit u S:S→βSu_S: S \to \beta S may also be denoted prin Sprin_S, as it takes an element x∈Sx \in S to the principal ultrafilter

prin S(x)={U⊆S:x∈U}prin_S(x) = \{U \subseteq S: x \in U\}

and now the unit condition says prin S(x)⇝ ξxprin_S(x) \rightsquigarrow_\xi x for all xx. For ξ=conv(𝒪)\xi = conv(\mathcal{O}), this says N x⊆prin S(x)N_x \subseteq prin_S(x), or that x∈Vx \in V for all neighborhoods V∈N xV \in N_x, which is a tautology.

One of our goals is to prove the following theorem:

Ultrafilter monad on RelRel

Before rolling up our sleeves and proving the main theorem, we pause to consider some more abstract contexts in which to place the concept of lax β\beta-module, leading up to the context of generalized multicategories.

Extending the ultrafilter functor to RelRel

First we examine more closely the extension of the ultrafilter functor β:Set→Set\beta: Set \to Set to RelRel, showing in particular that the extension is a strict functor. First we slightly rephrase our earlier definition:

Any span of functions (h:S→X,k:S→Y)(h: S \to X, k: S \to Y) that represents rr (in the sense that r=kh or = k h^{o} in the bicategory of relations) would serve in place of (f,g)(f, g), since for any such span there is an epi s:S→Rs: S \to R with h=fsh = f s, k=gsk = g s, whence β(s)\beta(s) is epi (because the epi ss splits in SetSet) and we have

β(g)β(f) o = β(g)β(s)β(s) oβ(f) o = β(g)β(s)(β(f)β(s)) o = β(gs)β(fs) o = β(k)β(h) o.\array{ \beta(g)\beta(f)^{o} & = & \beta(g)\beta(s)\beta(s)^{o}\beta(f)^{o} \\ & = & \beta(g)\beta(s)(\beta(f)\beta(s))^{o} \\ & = & \beta(g s)\beta(f s)^{o} \\ & = & \beta(k)\beta(h)^{o}. }

In particular, β¯\bar{\beta} is well-defined. Since β¯\bar{\beta} extends β:Set→Set\beta: Set \to Set, there is no harm in writing β(r)\beta(r) in place of β¯(r)\bar{\beta}(r). If r≤r′:X→Yr \leq r': X \to Y, then β(r)≤β(r′)\beta(r) \leq \beta(r') (as can be seen from the calculation displayed above, but replacing the epi ss by a general map tt, and the first equation by an inequality ≥\geq).

Ultrafilter monad on the equipment Rel\mathbf{Rel}

As mentioned in an earlier section, the natural transformations u=prin:1 Set→βu = prin: 1_{Set} \to \beta, m:ββ→βm: \beta\beta \to \beta do not extend to (strict) natural transformations on the locally posetal bicategory RelRel, but only to transformations that are op-lax in the sense of inequalities

prin Y∘r≤β(r)∘prin X,m Y∘ββ(r)≤β(r)∘m Xprin_Y \circ r \leq \beta(r) \circ prin_X, \qquad m_Y \circ \beta\beta(r) \leq \beta(r) \circ m_X

for every relation r:X→Yr: X \to Y. These are equivalent to inequalities

r≤prin Y o∘β(r)∘prin X,ββ(r)≤m Y o∘β(r)∘m Xr \leq prin_Y^o \circ \beta(r) \circ prin_X, \qquad \beta\beta(r) \leq m_Y^o \circ \beta(r) \circ m_X

and they may be deduced simply by staring at naturality diagrams in SetSet, in which we represent or tabulate rr by π 2∘π 1 o\pi_2 \circ \pi_1^o:

R ββ(R) π 1↙ ↓ prin R ↘π 2 ββπ 1↙ ↓ m R ↘ ββπ 2 X β(R) Y ββ(X) β(R) ββ(Y) prin X↓ ↙ βπ 1 βπ 2↘ ↓ prin Y m X↓ ↙ βπ 1 βπ 2↘ ↓m Y β(X) β(Y) β(X) β(Y)\array{ & & R & & & & & & & & & & \beta\beta (R) & & \\ & \mathllap{\pi_1} \swarrow & \downarrow_\mathrlap{prin_R} & \searrow \mathrlap{\pi_2} & & & & & & & & _\mathllap{\beta\beta\pi_1} \swarrow & \downarrow_\mathrlap{m_R} & \searrow_\mathrlap{\beta\beta\pi_2} & \\ X & & \beta (R) & & Y & & & & & & \beta \beta (X) & & \beta (R) & & \beta \beta (Y) \\ _\mathllap{prin_X} \downarrow & \swarrow_\mathrlap{\beta \pi_1} & & _\mathllap{\beta \pi_2} \searrow & \downarrow_\mathrlap{prin_Y} & & & & & & \mathllap{m_X} \downarrow & \swarrow_\mathrlap{\beta \pi_1} & & _\mathllap{\beta \pi_2} \searrow & \downarrow \mathrlap{m_Y} \\ \beta (X) & & & & \beta (Y) & & & & & & \beta (X) & & & & \beta (Y) }

To get an actual monad, it is more satisfactory in this context to consider not the bicategory RelRel, but rather the equipment or framed bicategory Rel\mathbf{Rel}. That is, there is a 2-category EquipEquip of equipments (as a sub-2-category of a 2-category of double categories), so that the notion of monad makes sense therein, and it turns out the data to hand induces such a monad β¯:Rel→Rel\bar{\beta}: \mathbf{Rel} \to \mathbf{Rel}.

In more detail: the 0-cells of Rel\mathbf{Rel} are sets, and the horizontal arrows are relations between sets. Vertical arrows are functions between sets, and a 2-cell of shape

A →r B f↓ ⇓ ↓g C →s D;\array{ A & \stackrel{r}{\to} & B \\ \mathllap{f} \downarrow & \Downarrow & \downarrow \mathrlap{g} \\ C & \stackrel{s}{\to} & D; }

is an inequality g∘r≤s∘fg \circ r \leq s \circ f. We straightforwardly get a double category Rel\mathbf{Rel}, and the ultrafilter functor on SetSet extends to a functor β¯:Rel→Rel\bar{\beta}: \mathbf{Rel} \to \mathbf{Rel} between double categories (or in this case, equipments), preserving all structure in sight.

Some attention must be paid to the notion of transformation between functors F,G:B→CF, G: \mathbf{B} \to \mathbf{C} between equipments. A transformation η:F→G\eta: F \to G assigns to each 0-cell bb of B\mathbf{B} a vertical arrow ηb:Fb→Gb\eta b: F b \to G b, and to each horizontal arrow r:b→b′r: b \to b' a 2-cell ηr\eta r of the form

Fb →Fr Fb′ ηb↓ ⇓ηr ↓ηb′ Gb →Gr Gb′;\array{ F b & \stackrel{F r}{\to} & F b' \\ \mathllap{\eta b} \downarrow & \Downarrow \mathrlap{\eta r} & \downarrow \mathrlap{\eta b'} \\ G b & \stackrel{G r}{\to} & G b'; }

suitably compatible with the double category structures.

We thus find that the op-lax structures of the transformations prin:1→β¯prin: 1 \to \bar{\beta}, m:β¯β¯→β¯m: \bar{\beta} \bar{\beta} \to \bar{\beta} on RelRel qua bicategory are exactly what we need to produce honest transformations u:1→β¯u: 1 \to \bar{\beta}, m:β¯β¯→β¯m: \bar{\beta} \bar{\beta} \to \bar{\beta} on Rel\mathbf{Rel} qua equipment, and the result is an ultrafunctor monad on the equipment Rel\mathbf{Rel}.

Given a monad TT on an equipment B\mathbf{B}, one may proceed to construct a horizontal Kleisli equipment HKl(B,T)HKl(\mathbf{B}, T) with the same 0-cells and vertical arrows as B\mathbf{B}, but whose horizontal arrows are of the form r:b→Tb′r: b \to T b'. A 2-cell in HKl(B,T)HKl(\mathbf{B}, T) (with vertical source ff and vertical target gg) is a 2-cell in B\mathbf{B} of the form

b →r Tb′ f↓ ⇓α ↓Tg c →s Tc′;\array{ b & \stackrel{r}{\to} & T b' \\ \mathllap{f} \downarrow & \Downarrow \mathrlap{\alpha} & \downarrow \mathrlap{T g} \\ c & \stackrel{s}{\to} & T c'; }

with horizontal compositions being performed in familiar Kleisli fashion. (When we say “familiar Kleisli fashion”, we are using the fact that an equipment allows one to “translate” vertical arrows, in particular the map m b:TTb→Tbm_b: T T b \to T b, into horizontal arrows, which are then composed horizontally. Similarly, the unit of the monad is translated into a horizontal arrow, where it plays the role of an identity in the Kleisli construction.)

In an equipment, there is a notion of monoid and monoid homomorphism. A monoid consists of a horizontal arrow ξ:b→b\xi: b \to b together with unit and multiplication 2-cells

b →1 b b b →ξb→ξ b 1↓ ⇓η ↓1 1↓ ⇓μ ↓1 b →ξ b b →ξ b\array{ b & \stackrel{1_b}{\to} & b & & & & & & b & \stackrel{\xi}{\to}\;\;\; b \;\;\; \stackrel{\xi}{\to} & b\\ \mathllap{1} \downarrow & \Downarrow \mathrlap{\eta} & \downarrow \mathrlap{1} & & & & & & \mathllap{1} \downarrow & \Downarrow \mathrlap{\mu} & \downarrow \mathrlap{1}\\ b & \stackrel{\xi}{\to} & b & & & & & & b & \stackrel{\xi}{\to} & b }

satisfying evident identities. A monoid homomorphism from (b,ξ)(b, \xi) to (c,θ)(c, \theta) consists of a vertical arrow and 2-cell (f,ϕ)(f, \phi) of the form

b →ξ b f↓ ⇓ϕ ↓f c →θ c\array{ b & \stackrel{\xi}{\to} & b \\ \mathllap{f} \downarrow & \Downarrow \mathrlap{\phi} & \downarrow \mathrlap{f} \\ c & \underset{\theta}{\to} & c }

that is suitably compatible with the unit and multiplication cells.

The following notion gives an interim notion of generalized multicategory that applies in particular to relational β\beta-modules.

Proof of Main Theorem

We now return to the task of proving theorem .

We now break up our Main Theorem into the following two theorems.

The next theorem establishes the converse of the preceding theorem; the two theorems together establish the Main Theorem. First we need a remark and a lemma.

This completes the proof of the Main Theorem (theorem ).

Continuous maps

Properties

As above, a subset AA of SS is open if A∈𝒰A \in \mathcal{U} whenever 𝒰→x∈A\mathcal{U} \to x \in A. On the other hand, by lemma , AA is closed if x∈Ax \in A whenever A∈𝒰→xA \in \mathcal{U} \to x.

A relational β\beta-module is compact if every ultrafilter converges to at least one point. It is Hausdorff if every ultrafilter converges to at most one point. Thus, a compactum is (assuming UFUF) precisely a relational β\beta-module in which every ultrafilter converges to exactly one point, that is in which the action of the monad β\beta lives in SetSet rather than in RelRel. Full proofs may be found at compactum; see also ultrafilter monad.

A continuous map ff from (X,ξ:βX→X)(X, \xi: \beta X \to X) to (Y,θ:βY→Y)(Y, \theta: \beta Y \to Y) is proper if the square

βX →ξ X β(f)↓ ↓f βY →θ Y\array{ \beta X & \stackrel{\xi}{\to} & X \\ \mathllap{\beta (f)} \downarrow & & \downarrow \mathrlap{f} \\ \beta Y & \underset{\theta}{\to} & Y }

commutes (strictly) in RelRel, and ff is open if the square

βX →ξ X β(f) o↑ ↑f o βY →θ Y\array{ \beta X & \stackrel{\xi}{\to} & X \\ \mathllap{\beta (f)^o} \uparrow & & \uparrow \mathrlap{f^o} \\ \beta Y & \underset{\theta}{\to} & Y }

commutes in RelRel. From this point of view, a space XX is Hausdorff if the diagonal map δ:X→X×X\delta: X \to X \times X is open, and compact if ϵ:X→1\epsilon: X \to 1 is proper (and these facts remain true even for pseudotopological spaces). See Clementino, Hofmann, and Janelidze, infra corollary 2.5.

The following ultrafilter interpolation result is due to Pisani:

For an convergence-approach extension of this result to exponentiable maps in TopTop, see Clementino, Hofmann, and Tholen.

For more on this, see Clementino, Hofmann, and Janelidze.

Relation to nonstandard analysis

In nonstandard analysis (which implicitly relies throughout on UFUF), one may define a topological space using a relation between hyperpoints (elements of S *S^*) and standard points (elements of SS). If uu is a hyperpoint and xx is a standard point, then we write u≈xu \approx x and say that xx is a standard part? of uu or that uu belongs to the halo (or monad, but not the category-theoretic kind) of xx. This relation must satisfy a condition analogous to the condition in the definition of a relational β\beta-module. The nonstandard defintions of open set, compact space, etc are also analogous. (Accordingly, one can speak of the standard part of uu only for Hausdorff spaces.)

So ultrafilters behave very much like hyperpoints. This is not to say that ultrafilters are (or even can be) hyperpoints, as they don't obey the transfer principle. Nevertheless, one does use ultrafilters to construct the models of nonstandard analysis in which hyperpoints actually live. Intuitions developed for nonstandard analysis can profitably be applied to ultrafilters, but the transfer principle is not valid in proofs.

Relation to other topological concepts

If β\beta is treated as a monad on SetSet instead of on RelRel, then its algebras are the compacta (the compact Hausdorff spaces), again assuming UPUP; see ultrafilter monad, and more especially compactum.

One might hope that there would be an analogous treatment of uniform spaces based on an equivalence relation between ultrafilters. (In nonstandard analysis, this becomes a relation ≈\approx of infinite closeness between arbitrary hyperpoints, instead of only a relation between hyperpoints and standard points.) The description in terms of generalized multicategories is known to generalize to a description of uniform spaces, but rather than using relations between ultrafilters, this description uses pro-relations between points.

For more on relations between Barr’s approach to topological spaces, Lawvere’s approach to metric spaces, as well as uniform structures, prometric spaces, and approach structures, see Clementino, Hofmann, and Tholen.

References

• Michael Barr, Relational algebras, Springer Lecture Notes in Math. 137 (1970), 39-55. (pdf)

• Maria Manuel Clementino, Dirk Hofmann, and George Janelidze, On exponentiability of étale algebraic homomorphisms, Preprint 11-35, University of Coimbra. (pdf)

• Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, The convergence approach to exponentiable maps, Portugaliae Mathematica (Nova Series), Vol. 60 Issue 2 (2003), 139-160. (web) (pdf)

• Maria Manuel Clementino, Dirk Hofmann, and Walter Tholen, One Setting for All: Metric, Topology, Uniformity, Approach Structure. (Springer Link, doi: 10.1023/B:APCS.0000018144.87456.10)

• Gavin J. Seal, Canonical and op-canonical lax algebras, Theory and Applications of Categories, 14 (2005), 221–243. (web)

• Claudio Pisani, Convergence in exponentiable spaces, Theory Appl. Categories 5 (1999), 148-162. (web)

• Geoff Cruttwell and Mike Shulman, A unified framework for generalized multicategories, arxiv/0907.2460